
Binary cross-entropy (BCE)
\(BCE = -\frac{1}{N} \sum _{i=1}^{N}g_{i} \cdot log (p_{i}) + (1 - g_{i})\cdot log(1 - p_{i}) \) 
where the sums are calculated over the N voxels of the predicted mask volume (softmax output of the network) \(p_i\)  \(\in \)  P and the ground truth binary mask volume \(g_i\)  \(\in \)  G.

Weighted cross-entropy (WCE)
\(WCE = -\frac{1}{N} \sum _{i=1}^{N} w_{k}\cdot (g_{i} \cdot log (p_{i}) + (1 - g_{i})\cdot log(1 - p_{i})) \) 
where \(w_{k}\)  is a vector with one multiplicative coefficient per class \(k\) . Values smaller than 1 reduce, while values greater 1 increase the weighting of this class. This loss was introduced by Dalca et al.[1]}.

Jaccard distance loss (JDL)
\(JDL = 1 - \frac{\sum _{i=1}^{N} g_{i} \cdot p_{i} + smooth}{\sum _{i=1}^{N} g_{i} + \sum _{i=1}^{N} p_{i} - \sum _{i=1}^{N} g_{i} \cdot p_{i} + smooth}\) 
with \(smooth\)  set to 1 to avoid dividing by zero.

Soft Dice loss (SDL)
\(SDL = 1 - \frac{1}{C} \sum _{i=1}^{C} DSC_{i}\) 
where C represents the foreground classes and \(DSC_{i}\)  the binary soft Dice coefficient (eq. REF ) for one class. The original SDL by Milletari et al.[2]} used the squared sum, which did not work well in our experiments. Because of this, the plain sum, as described by Drozdzal et al. [3]}, is used.

